Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Lamberti, Paola

doi:10.1007/s10543-009-0237-9

Cited by 19 publications

(15 citation statements)

References 22 publications

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“…and it has been used in many applications (see Lamberti 2008, 2007;Dagnino et al 2012Dagnino et al , 2013Lamberti 2009;Sablonnière 2003a,b;Wang 2001; Li 2004a and references therein). This paper wants also to be a further contribution to the researches on the surface construction based on above blending functions spanning S 1 2 (T mn ).…”

Section: Introductionmentioning

confidence: 99%

Curve network interpolation by C 1 quadratic B-spline surfaces

Dagnino

Lamberti

Remogna

2015

Computer Aided Geometric Design

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Please cite this article in press as: Dagnino, C., et al. Curve network interpolation by C 1 quadratic B-spline surfaces. Comput. Aided Geom. Des. (2015), http://dx. Highlights• Interpolation of a B-spline curve network by a surface based on C 1 quadratic B-splines on criss-cross triangulations.• Proof of the existence and uniqueness of the surface and constructive algorithm for its generation.• Numerical and graphical results and comparisons with other spline methods. AbstractIn this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C 1 quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.

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Section: Introductionmentioning

confidence: 99%

Curve network interpolation by C 1 quadratic B-spline surfaces

Dagnino

Lamberti

Remogna

2015

Computer Aided Geometric Design

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“…In [25] the problem of efficient evaluation of box splines is addressed by making use of the local Bernstein representation of basis functions on each triangle. Also, numerical integration schemes, which are important for applications, based on quasi-interpolation have been considered in [6,29]. Recent applications of box splines include surface fitting [23], and solving linear elasticity problems in isogeometric analysis [19].…”

Section: Introductionmentioning

confidence: 99%

Completeness characterization of Type-I box splines

Villamizar¹,

Mantzaflaris²,

Jüttler³

2020

Preprint

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We present a completeness characterization of box splines on three-directional triangulations, also called Type-I box spline spaces, based on edge-contact smoothness properties. For any given Type-I box spline, of specific maximum degree and order of global smoothness, our results allow to identify the local linear subspace of polynomials spanned by the box spline translates. We use the global super-smoothness properties of box splines as well as the additional super-smoothness conditions at edges to characterize the spline space spanned by the box spline translates. Subsequently, we prove the completeness of this space space with respect to the local polynomial space induced by the box spline translates. The completeness property allows the construction of hierarchical spaces spanned by the translates of box splines for any polynomial degree on multilevel Type-I grids. We provide a basis for these hierarchical box spline spaces under explicit geometric conditions of the domain.

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“…Such integration is inefficient as it ignores the continuity of the spline functions across macro-and micro-elements. In a related line of research, work has been done on integration schemes based on quasi-interpolants over (refined) triangulations [12,25,31]; see also the survey [26]. Recently, we have shown that the quadrature rule of Hammer and Stroud [16] for cubic polynomials is exact for a larger space of functions, namely the C 1 cubic Clough-Tocher spline space over a macro-triangle, if and only if the split-point is the barycentre of the macro-triangle [21].…”

Section: Introductionmentioning

confidence: 99%

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

Bartoň

Kosinka

2019

Journal of Computational and Applied Mathematics

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The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the C 1 cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C 1 quadratic Powell-Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the C 1 quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.

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Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Cited by 19 publications

References 22 publications

Curve network interpolation by C 1 quadratic B-spline surfaces

Curve network interpolation by C 1 quadratic B-spline surfaces

Completeness characterization of Type-I box splines

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

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